Abstract
In this paper, the following [Formula: see text]-dimensional distribution-dependent stochastic differential equation driven by a pure jump process is studied: [Formula: see text] where [Formula: see text] denotes the distribution of [Formula: see text]. The differentiability of the map [Formula: see text] is investigated in the sense of [Formula: see text]. By the Malliavin calculus for jump processes, the following Bismut type derivative formula is established, [Formula: see text] where [Formula: see text] is a test function and [Formula: see text] is a random variable depending on the initial value [Formula: see text]. Sharp gradient estimates in short time are also obtained.
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